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\date{\today}
\title{Present Certainty Equivalent Evaluation of Financial Hedging vs. Operational Hedging}

\begin{document}
\doublespacing

\maketitle
%\textbf{Present Certainty Equivalent Evaluation of Financial Hedging vs. Operational Hedging}
\bigskip





\section{Introduction}

\emph{ One could argue that all firms should make decision based on risk-neutral measures. As widely observed and highlighted by examples above, firms take uncertainty into account in both short and long term decisions Smith and Nau 1995 describe in detail the rational behind risk-sensitive decision making, they note that the decision makers are firm's managers, the managers have their own risk profiles, thus even the large public company are risk-sensitive.}

Other reasons for risk aversion and impropriety of NPV include \\
(1) incomplete market\\ 
(2) consumers may not diversify their risks\\
(3) existence of other costs related to the variation of cash flows, including bankruptcy cost, etc..


\section{Literature Review}

Smith 1998 (page 1698) categorizes  the elements of Certainty Equivalence as risk premium and delay premium. Note that utility of NPV method used by Bouakiz and Sobel (1992) ignores the delay premium.


Smith and Nau (1995) show that for complete market (every risk is tradable), the following three methods, NPV, real option, and decision analysis are equivalent. That is, the optimal decisions, optimal trading strategies, and optimal financing strategies are the same. This is because tradability allows for risk neutral treatment. They note, however, that in case of non-tradable risk, like the demand in our case,  NPV and real options approaches cannot be naively applied, but instead, use of decision analysis with integrated evaluation is more appropriate. What Smith and Nau label as decision analysis is actually identical in spirit to our approach and can be labeled as a special case of Epstein-Zin framework (to be verified???).

Fama (1970) uses time additive utility functions to study multi-period optimal consumption decisions.  Blanchard and Mankiw (1988) use the same additive utility function to study certainty equivalence in consumption. They use both quadratic and exponential utility functions (as examples?).

\section{Basic Model \label{sect:basicModel}}


Our basic model differs from some of the risk-averse literature in that we explicitly consider the beliefs and preferences of the MRN. We attribute beliefs and preferences to this MRN, as if it was privately owned and operated by a single owner/manager (note adding references). We will discuss the implications of this assumption in the conclusion section.

Similarly to Smith and Nau (1995), the MRN's beliefs are captured by his subjective probability of the random sources, while his preference is captured by a non-decreasing concave utility function. MRN's goal is to maximize his expected utility for consumptions. Smith and Nau use an additive exponential utility function to represent the decision maker's preferences.


The cash consumption takes place at the end of each period. The cash consumed is  a result of (a) operation cash flow and (b) cash flow from security trading. The operational cash flow is affected by two decisions: the up-front allocation of capacity and production and transshipment quantities in each period.  The security trading cash flow is the result of the MRN's trading of securities in the financial market.

The financial market has two traded securities, a risk-free bond and a foreign currency. To simplify the notation and analysis, we assume that the foreign currency and bond pay the same dividend rate $r$.  (Note: can we generalize to a different dividend rate in foreign currency???). The MRN's trading strategies are defined in space of  borrowing and lending the risk-free bond and the foreign currency in each period. In practice, while all MRNs do borrow and lend risk-free bonds, not all of them engage in trading of risky foreign currency in order to hedge the exchange rate risk. That is, not all are involved in financial hedging.


Once the operational decisions and trading strategies are decided, the randomness comes from two sources: the product demand and the exchange rate uncertainty. We express the exchange rate as the value of the foreign currency in domestic currency. Without loss of generality, the random demand is assumed to be independent of the exchange rate because any correlation can be decomposed into a perfectly correlated source and an independent source. The exchange rate is modeled as a Markovian process, consistent with the model in Hull (1998).


Smith (1998) shows that without financial hedging, but with optimal trading in the risk-free bond, maximizing MRN's expected consumption utility is equivalent to the maximizing  MRN's Present Certainty Equivalent Value in a general setting. We proceed to show that it is true in our setting without financial hedging.

\section{Present Certainty Equivalent Valuation without Financial Hedging}


\subsection{Definition of Present Certainty Equivalent Value}

The Present Certainty Equivalent Value of an uncertain cash flow is a deterministic cash value in the initial period such that  the maximum expected utility generated in all periods by it in combination with bond investment, across all possible bond instruments, is the same as the utility generated by the uncertain cash flow with bond investment, also across all possible bond investments. To operationalize this definition, we use he following notation:
\begin{itemize}
\item $n\in \{0,1, \cdots, N\}$: period index
\item $X_n$: period $n$ uncertain cash flow that represents the decision maker's private belief
\item $x_n$: realization of $X_n$
\item $\beta_n$:  cumulative shares of risk-free bond held in period $n$, after any purchases of additional bonds in period $n$
\item $U_n$: utility function that evaulates the decision maker's preferences for consumptions in periods $n$ to $N$.
\item $\mathcal{U}_n$: the maximum expected utility generated by the cash flows combined with  bond investments for periods $n$ to $N$.
\item $PCEV_n$: the present certainty equivalent value in period $n$ for  cash flows in period $n$ to $N$.
\end{itemize}
Note that the cumulative bond investment, $\beta_n$, is a strategy that depends on the realization of cash flows in the past. Furthermore $\beta_N=0$ implies that there is no consumption after period $N$ and all the borrowing has to be repaid in full in period N. In each period, the sequence of events is that
\begin{itemize}
\item[0] Any uncertainty for period $n$ is resolved.
\item[1] MRN receives income cash.
\item[2] MRN enters financial trade.
\item[3] MRN receives or pay out trade cash.
\item[4] MRN consumes the net cash.
\end{itemize}


%\begin{figure}[htbp]
%\begin{center}
%\includegraphics[scale=1]{events.pdf}
%\caption{The sequence of events in each period}
%\label{fig:events}
%\end{center}
%\end{figure}


For clarity of the exposition we first assume that the risk-free bond pays $0$ dividend if held for any period of time. This implies that the price of the bond is $1$ is each period.

\begin{definition} The maximum expected utility for future uncertain cash flows and a given initial bond holding of $\beta_{n-1}$ that carries over from the past, is defined as follows:
\begin{align} \label{eqn:maxUti}
    &\mathcal{U}_n(X_n,\cdots, X_{N}|\beta_{n-1}) \nonumber \\
    & = \max_{\beta_n, \cdots, \beta_{N-1}} \E [U_n(X_n+\beta_{n-1}-\beta_n, \cdots, X_i + \beta_{i-1}-\beta_i, \cdots, X_N + \beta_{N-1}) ]
\end{align}
\end{definition}

\begin{definition} The present certainty equivalent value (PCEV) is defined as a constant such that,
\begin{align} \label{eqn:PCEV}
    \mathcal{U}_n(PCEV_n(X_n, \cdots, X_N|\beta_{n-1}), 0,\cdots,0|\beta_{n-1})=\mathcal{U}_n(X_n,\cdots, X_{N}|\beta_{n-1})
\end{align}
\end{definition}
In general, calculating PCEV based on the above two definitions may be not easy. Therefore, we first assume that  \\
(a) Utility function is time additive
\begin{align}\label{eqn:addUti}
 U_n(x_n, \cdots, x_{N}) = \sum_{i = n}^Nu(x_n)
\end{align}
Applying that,  (\ref{eqn:maxUti}) becomes:
\begin{align} \label{eqn:maxAddUti}
    \mathcal{U}_n(X_n,\cdots, X_{N}|\beta_{n-1}) = \max_{\beta_n, \cdots, \beta_{N-1},\beta_N = 0} \sum_{i=n}^N  \E [ u( X_i + \beta_{i-1}-\beta_i)]
\end{align}
(b) Utility function has constant absolute risk aversion, that is, it is exponential
\begin{align}\label{eqn:expUti}
u(x) = -\exp(-x)\end{align}

For exponential utility function, the following are well-known properties and will be used repeatedly. 
\begin{proposition} \label{prop:exp}
The exponential utility function $u$ satisfies: \\
(i) $u(x+y) = -u(x)u(y)$ \\
(ii) The derivative $u' (x) = -u(x)$ \\
(iii)The inverse $u^{-1}(-xy) = u^{-1} (x) + u^{-1} (y)$
\end{proposition}

To study PCEV of a stream of uncertain cash flows,  we first consider properties of an uncertain cash flow in one  period.
\begin{lemma} \label{lem:randomX_n}
If the utility function is time additive, concave and stationary, then
\begin{align}
 \mathcal{U}_n(X_n,0, \cdots, 0|\beta_{n-1}) &=(N-n+1) \E_{X_n} [u(\frac{X_n+\beta_{n-1}}{N-n+1})]
\end{align}
The corresponding optimal bond shares held in each periods are
\begin{align} \label{eqn:beta_iOpt}
\beta_i = \frac{(N-i)(X_n+ \beta_{n-1})}{N-n+1} \quad \forall i \in \{n, \cdots, N-1\}
\end{align}
\end{lemma}
\proof According to the sequence of events, the decision of $\beta_i $ $(\forall i \geq n) $ is made after the realization of income cash $X_n$. Thus,  the maximum utility problem becomes
\begin{align*}
\E\big{[}\max_{\beta_n, \cdots, \beta_{N-1}, \beta_N = 0} \{u(X_n +\beta_{n-1} - \beta_n) + \sum_{i=n+1}^N u(\beta_i - \beta_{i+1})\}|X_n\big{]}
\end{align*}
Concavity implies that an non-equal values can be equalized by their average. 
\endproof

Intuitively if the decision maker has some income cash in the initial period, then  consuming this cash in equal installments in all periods maximizes his total consumption utility. Since this intuition holds true for the constant cash, e.g., $PCEV_n$, in the initial period, we have

\begin{lemma} \label{lem:pcev-Xn}
The $PCEV_n$ of an uncertain cash in initial period $n$ is
\begin{align} \label{eqn:pcev-Xn}
PCEV_n(X_n,0,\cdots, 0|\beta_{n-1}) = (N-n+1)u^{-1}\big{(} \E [u(\frac{X_n}{N-n+1})]\big{)} 
\end{align}
and the $PCEV_n$ is independent of initial wealth $\beta_{n-1}$.
\end{lemma}
\proof
As a special case of Lemma~\ref{lem:randomX_n}, the maximum utility of  $PCEV_n$ in the initial period $n$ is
\begin{align*} 
    \mathcal{U}_n(PCEV_n, 0,\cdots,0|\beta_{n-1}) = (N-n+1)u(\frac{PCEV_n+\beta_{n-1}}{N-n+1})
\end{align*}
From (\ref{eqn:PCEV}), we have
\begin{align*}
u(\frac{PCEV_n + \beta_{n-1}}{N-n+1})   &=  \E[u(\frac{X_n + \beta_{n-1}}{N-n+1})]\\
-u(\frac{PCEV_n}{N-n+1})u(\frac{\beta_{n-1}}{N-n+1}) &= -\E [u(\frac{X_n}{N-n+1})u(\frac{\beta_{n-1}}{N-n+1})]
\end{align*}
The last equality follows from (i) in Proposition~\ref{prop:exp}.  Finally because $\beta_{n-1}$ is deterministic, canceling its utility in both sides of the equation concludes the proof.
\endproof



Based on Lemma~\ref{lem:pcev-Xn}, we will write $PCEV_n$ without conditioning on $\beta_{n-1}$.
This Lemma and the utility function Proposition~\ref{prop:exp}  (i) and (iii) immediate imply the following:
\begin{lemma}\label{lem:delta}
If $\delta$ is a deterministic cash flow, then
\[PCEV_n(X_n+\delta,0,\cdots,0) = PCEV_n(X_n,0,\cdots,0) + \delta\]
\end{lemma}

With these properties, we are ready to generalize the independence property to multiple future uncertain cash flows and to develop a recursive procedure for computing its PCEV.

\begin{lemma}\label{lemma:indCF}
If the cash flow in each period is independent, then $PCEV_n$ is independent of $\beta_{n-1}$ and can be computed recursively, that is,
\begin{align}
&PCEV_n(X_n,\cdots,X_N|\beta_{n-1}) = PCEV_n(X_n,\cdots, X_N) \label{eqn:pcev-ind} \\
&PCEV_n(X_n, \cdots, X_N|\beta_{n-1}) = PCEV_n(X_n, 0, \cdots, 0) + PCEV_{n+1} (X_{n+1}, \cdots, X_N) \label{eqn:pcev-recur}
\end{align}
\end{lemma}

\proof The Lemma is true when $n=N$. Suppose it holds a given $n
\leq N$. We will show that both equations (\ref{eqn:pcev-ind}) and (\ref{eqn:pcev-recur}) hold for $n-1$. We start with (\ref{eqn:pcev-recur}), which becomes:
\[PCEV_{n-1}(X_{n-1},X_n, \cdots, X_N|\beta_{n-2}) = PCEV_{n-1}(X_{n-1},0,\cdots, 0) + PCEV_n(X_n, \cdots, X_N)\]
It is sufficient to show that the maximum utilities generated by the two sides of the above equation are equal. We start with the left side. By ~(\ref{eqn:maxUti}) for $n-1$, we have:
\begin{eqnarray*}
&&\mathcal{U}_{n-1}(PCEV_{n-1}(X_{n-1},\cdots,X_N|\beta_{n-2}),0,\dots,0 |\beta_{n-2})  \\&&= \mathcal{U}_{n-1}(X_{n-1},\cdots,X_N|\beta_{n-2}) = \max_{  \beta_{n-1}, \cdots, \beta_{N-1},\beta_N=0} \E[\sum_{i=n-1}^N u(X_i + \beta_{i-1}-\beta_i)] \\
&& = \max_{\beta_{n-1}} \{\E_{X_{n-1}}[ u(X_{n-1}+\beta_{n-2}-\beta_{n-1})+ \max_{\beta_{n}, \cdots,\beta_{N-1}, \beta_N=0 }\{ \E [\sum_{i=n}^N u(X_i + \beta_{i-1} - \beta_i)]\}]\} \\
&& = \max_{\beta_{n-1}} \{\E_{X_{n-1}}[ u(X_{n-1}+\beta_{n-2}-\beta_{n-1})+   \mathcal{U}_n(X_n,\cdots, X_N |\beta_{n-1}) ]\} \quad  \text{by ~(\ref{eqn:maxUti})}\\
&& =\max_{\beta_{n-1}} \{\E_{X_{n-1}}[ u(X_{n-1}+\beta_{n-2}-\beta_{n-1})+ \mathcal{U}_n(PCEV_n(X_n, \cdots, X_N),0,\cdots,0)|\beta_{n-1} )]\}\\
&&= \max_{\beta_{n-1}, \cdots,\beta_{N-1}, \beta_N=0 } \{\E[u(X_{n-1}+\beta_{n-2} - \beta_{n-1}) \\&& + u(PCEV_n(X_n, \cdots, X_N) + \beta_{n-1}-\beta_n) + \sum_{i=n+1}^N u( \beta_{i-1} - \beta_i)]\}
\end{eqnarray*}
where the last equality follows from ~(\ref{eqn:maxUti}) again and the second to the last equality from ~(\ref{eqn:PCEV}) and induction assumption that $PCEV_n$ is independent of $\beta_{n-1}$. Using variable substitution $\beta'_{n-1} = \beta_{n-1} + PCEV_n(X_n, \cdots, X_N)$,  the above expression becomes
\begin{eqnarray*}
&=& \max_{ \beta'_{n-1}, \cdots, \beta_{N-1},  \beta_N=0}\{ \E[u(X_{n-1}+ PCEV_{n}(X_n,\cdots,X_N) +\beta_{n-2}- \beta'_{n-1})\\
&&+ u(\beta'_{n-1}-\beta_n) + \sum_{i=n+1}^N u(\beta_{i-1}-\beta_i)] \} \\
&=& \mathcal{U}_{n-1}(X_{n-1}+PCEV_n(X_n,\cdots,X_N), 0, \cdots,0|\beta_{n-2}) \\
&=& \mathcal{U}_{n-1}(PCEV_{n-1}(X_{n-1},0,\cdots,0) + PCEV_n(X_n,\cdots,X_N) ,0\cdots,0|\beta_{n-2})
\end{eqnarray*}

The second to the last equality follows from ~(\ref{eqn:maxUti}).
The last equality follows from Lemma~\ref{lem:delta} because $PCEV_n$ is a deterministic cash flow, which concludes the induction proof of (\ref{eqn:pcev-recur}). Statement (\ref{eqn:pcev-ind}) for $n-1$ follows immediately from (\ref{eqn:pcev-recur}).
\endproof


This lemma immediately imply

\begin{theorem} \label{theo:indCash} 
If the uncertain cash flows are independent in all periods, then
\[PCEV_n(X_n, \cdots, X_N)  = \sum_{i=n}^N PCEV_i(X_i,0,\cdots,0)\]
\end{theorem}


\subsection{Markov-Modulated Cash Flow Example for Two Periods}

In this section we consider the cash flow to be a Markov Process. Since the future cash flow's distribution depends on the realization of the past cash flow, both the maximum utility and the PCEV depend on the past cash flow realization. Since the cash flow is a Markov process, these values depend only on the cash flow realization in the last period. We have to refine  utility definition ~(\ref{eqn:maxUti}) and PCEV definition~(\ref{eqn:PCEV}) to reflect the dependency on last cash flow, $x_{n-1}$.

The maximum utility becomes:
\begin{align} \label{eqn:maxUtiMK}
    &\mathcal{U}_n(X_n,\cdots, X_{N}|x_{n-1},\beta_{n-1}) \nonumber \\
    & = \max_{\beta_n, \cdots, \beta_{N-1}} \E [U_n(X_n+\beta_{n-1}-\beta_n, \cdots, X_i + \beta_{i-1}-\beta_i, \cdots, X_N + \beta_{N-1})|X_{n-1}=x_{n-1} ]
\end{align}

The present certainty equivalent value becomes:
\begin{align} \label{eqn:PCEVMK}
    \mathcal{U}_n(PCEV_n(X_n, \cdots, X_N|x_{n-1},\beta_{n-1}), 0,\cdots,0|\beta_{n-1})=\mathcal{U}_n(X_n,\cdots, X_{N}|x_{n-1},\beta_{n-1})
\end{align}


We first study a two-period problem to observe the structural properties that will be generalized to $N$ periods. Let $N=1$. We face two cash flows $X_0, X_1$, where $X_1$'s distribution is a function of $X_0$.

\subsubsection{$PCEV_1(X_1|x_0,\beta_0(x_0))$}
By definition~(\ref{eqn:maxUtiMK}), the maximum utility generated by PCEV is
\[\mathcal{U}_1(PCEV_1(X_1|x_0,\beta_0(x_0))|\beta_0(x_0)) = u(PCEV_1(X_1|x_0,\beta_0(x_0)) + \beta_0(x_0))\]
and similarly the maximum utility generated by the uncertain cash flow $X_1$ is,
\begin{eqnarray*}
\mathcal{U}_1(X_1|x_0,\beta_0(x_0)) &=& \E_{X_1}[u(X_1 + \beta_0(x_0))|X_0=x_0]\\
&=&\E_{X_1}[-u(X_1)u(\beta_0)|X_0=x_0]\\
&=&-\E_{X_1}[u(X_1)|X_0=x_0]u(\beta_0(x_0)|X_0=x_0)
\end{eqnarray*}
From ~(\ref{eqn:PCEVMK}), we have:
\[PCEV_1(X_1|x_0,\beta_0(x_0)) = u^{-1}(\E_{X_1}[u(X_1)|X_0=x_0]) \]
%Therefore the PCEV again does not depend on the bond investment carry-over from the past. and we have:
%\begin{align}\label{eqn:MKLast}
%PCEV_1(X_1|x_0) =u^{-1}(\E_{X_1}[u(X_1)|X_0=x_0])
%\end{align}
%\subsubsection{$PCEV_0(X_0, 0|\beta_{-1})$}

%Now we discuss a single cash flow at time 0 with initial wealth $\beta_{-1}$. Again by definition Eq.~(\ref{eqn:maxUtiMK}), the utility generated by the certain cash flow is:
%\begin{eqnarray*}
%\mathcal{U}_0(PCEV_0(X_0,0|\beta_{-1}),0|\beta_{-1}) &=& \max_{\beta_0}\{u(PCEV_0(X_0,0|\beta_{-1})+\beta_{-1}-\beta_0)+u(\beta_0)\}\\
%&=& 2u((PCEV_0(X_0,0|\beta_{-1})+ \beta_{-1})/2)
%\end{eqnarray*}
%Again by Eq.~(\ref{eqn:maxUtiMK}), the utility generated by the single uncertain cash flow is:
%\begin{eqnarray*}
%\mathcal{U}_0(X_0,0|\beta_{-1}) &=& \max_{\beta_0(X_0)} \{\E[u(X_0 + \beta_{-1}-\beta_0(X_0) + u(\beta_0(X_0)) ]\} \\
%&=& 2\E[u((X_0+\beta_{-1})/2)]\\
%&=& -2\E[u(X_0/2)]u(\beta_{-1}/2)
%\end{eqnarray*}
%The last inequality follows from independency of $\beta_{-1}$ on $X_0$ and the property (i) of the exponential utility function, and  the second to the last inequality from the first order optimality condition and property (ii) of the exponential utility function.

%By PCEV definition Eq.~(\ref{eqn:PCEVMK}), we can find the PCEV:
%\begin{eqnarray*}
%PCEV_0(X_0,0|\beta_{-1}) + \beta_{-1} &=& 2\{u^{-1}(-\E[u(X_0/2)]u(\beta_1/2))\}\\
%&=& 2u^{-1}(\E[u(X_0/2)]) + \beta_{-1} \text{by Property (iii)}
%\end{eqnarray*}
%Therefore we have the PCEV value independent of past investment (initial wealth) and specifically,
%\begin{align}\label{eqn:MKfirst}
%PCEV_0(X_0,0) = 2u^{-1}(\E[u(X_0/2)])
%\end{align}

%\subsubsection{$PCEV_0(X_0,X_1|\beta_{-1})$}
%Finally we study the PCEV of two uncertain cash flows with initial wealth. We start with definition Eq.~(\ref{eqn:maxUtiMK}) for PCEV,
%\begin{eqnarray*}
%\mathcal{U}_0(PCEV_0(X_0,X_1|\beta_{-1}),0|\beta_{-1}) &=& \max_{\beta_0}\{u(PCEV_0(X_0,X_1|\beta_{-1})+\beta_{-1}-\beta_0)+u(\beta_0)\}\\
%&=& 2u((PCEV_0(X_0,X_1|\beta_{-1})+ \beta_{-1})/2)
%\end{eqnarray*}
%Now we use definition Eq.~(\ref{eqn:maxUtiMK}) for the two uncertain cash flows,
% \begin{eqnarray*}
%\mathcal{U}_0(X_0,X_1|\beta_{-1}) &=& \max_{\beta_0(X_0)} \{\E_{X_0,X_1}[u(X_0 + \beta_{-1}-\beta_0(X_0) + u(X_1 + \beta_0(X_0)) ]\} \\
%&=& \max_{\beta_0(X_0)} \{\E_{X_0} [u(X_0+ \beta_{-1} - \beta_0(X_0)) + \E_{X_1}[u(X_1+\beta_0(X_0))|X_0]]\}\\
%& = & \max_{\beta_0(X_0)} \{\E_{X_0} [u(X_0+ \beta_{-1} - \beta_0(X_0)) + u(PCEV_1(X_1|X_0) + \beta_0(X_0))]\}\\
%&=& 2\E_{X_0}[u((X_0+PCEV_1(X_1|X_0) +\beta_{-1})/2)]\\
%&=& -2\E_{X_0}[u((X_0+PCEV_1(X_1|X_0)) /2)]u(\beta_{-1}/2)
%\end{eqnarray*}

%Using the PCEV definition Eq.~(\ref{eqn:PCEVMK}), we have:
%\[PCEV_0(X_0,X_1|\beta_{-1}) + \beta_{-1} = 2u^{-1}(\E_{X_0}[u((X_0+PCEV_1(X_1|X_0)) /2)]) + \beta_{-1} \]
%Thus we have the PCEV is independent of initial wealth, and we further have,
%\begin{align}
%PCEV_0(X_0,X_1) &=& 2u^{-1}(\E_{X_0}[u((X_0+PCEV_1(X_1|X_0)) /2)])\nonumber \\
%&=& PCEV_0(X_0 + PCEV_1(X_1|X_0),0)  \text{by Eq.~(\ref{eqn:MKfirst})}
%\end{align}


\subsection{PCEV of Markov Stochastic Cash Flow}
We show here ~(\ref{eqn:maxUtiMK}) and ~(\ref{eqn:PCEVMK}), and the time additive utility function imply,  first, that the PCEV value is independent of initial wealth, and second, that the PCEV value can be computed recursively. In order to study these properties of a stream of cash flows, we start with a single cash flow. Base on (\ref{eqn:pcev-Xn}), we have
\begin{lemma}\label{lem:oneCF-PCEV}
If utility function is time additive exponential, then
\begin{align}
&PCEV_n(X_n, 0,\cdots,0|x_{n-1},\beta_{n-1}) = (N-n+1)u^{-1}(\E_{X_n}[u(\frac{X_n}{N-n+1})|X_{n-1}=x_{n-1}]) \\
&PCEV_n(X_n,0,\cdots,0|x_{n-1}) = PCEV_n(X_n,0,\cdots,0|x_{n-1},\beta_{n-1})
\end{align}
\end{lemma}
%\proof For any cash flow $Y_n$ that may depend on $x_{n-1}$, we have by Eq.~(\ref{eqn:maxUtiMK}),
%\begin{align}
%\mathcal{U}_n(Y_n,0,\cdots,0|x_{n-1},\beta_{n-1}) =  -(N-n+1)\E_{Y_n}[u(\frac{Y_n}{N-n+1})|X_{n-1}=x_{n-1}]) u(\frac{\beta_{n-1}}{N-n+1})
%\end{align}
%The equality follows from the Property (i) and (ii) of the exponential utility function and the first order optimality condition.

%Since $Y_n$ is arbitrary, We can let $Y_n = PCEV_n(X_n,0,\cdots,0|x_{n-1},\beta_{n-1})$. Since it is a deterministic cash flow, the above expression is reduced to
%\[-(N-n+1)u(\frac{ PCEV_n(X_n,0,\cdots,0|x_{n-1},\beta_{n-1})}{N-n+1}) u(\frac{\beta_{n-1}}{N-n+1}) \]
%Now letting $Y_n = X_n$ and using Eq.~(\ref{eqn:PCEVMK}) imply the first statement. The second statement follows from the first statement immediately.
%\endproof

Now we are ready to study the PCEV of a stream of cash flows:

\begin{theorem}\label{theo:PCEVMK}
If the decision maker's preference is represented by the time additive utility function, then\\
(1) $PCEV_n$ is independent of $\beta_{n-1}$ and
\begin{align}
(2) &PCEV_n(X_n,\cdots,X_N|x_{n-1}, \beta_{n-1}) \nonumber \\&= PCEV_n(X_n + PCEV_{n+1}(X_{n+1},\cdots,X_N|X_n), 0,\cdots,0|x_{n-1},\beta_{n-1})
\end{align}
\end{theorem}
\proof
We prove above statements by induction. The statements hold for $n=N$ by previous Lemma~\ref{lem:oneCF-PCEV},  because it is a single cash flow case. Now suppose they hold for a given $n+1$. We first show statement (2) for $n$.

By (\ref{eqn:maxUtiMK}) and (\ref{eqn:PCEVMK}), the maximum utility generated by the PCEV in the left side of statement (2) is:
\begin{eqnarray*}
    &&\mathcal{U}_n(X_n,\cdots, X_{N}|x_{n-1},\beta_{n-1}) \nonumber \\
    & =& \max_{\beta_n(X_n), \cdots, \beta_{N-1}(X_{N-1})} \E [U_n(X_n+\beta_{n-1}-\beta_n, \cdots, X_i + \beta_{i-1}-\beta_i, \cdots, X_N + \beta_{N-1})|X_{n-1}=x_{n-1} ]\\
    &=&\max_{\beta_n(X_n), \cdots, \beta_{N-1}(X_{N-1}), \beta_N=0} \E [\sum_{i=n}^Nu(X_i + \beta_{i-1}-\beta_i)|X_{n-1}=x_{n-1}] \\
    &=& \max_{\beta_n(X_n)} \big{\{}\E_{X_n} [ \{u(X_n + \beta_{n-1} - \beta_{n})\\ && +\max_{\beta_{n+1}(X_{n+1}),  \cdots, \beta_{N-1}(X_{N-1}), \beta_N=0}\E [\sum_{i=n+1}^Nu(X_i + \beta_{i-1}-\beta_i)|X_{n}]\}|X_{n-1}=x_{n-1}]\big{\}} \\
    &=& \max_{\beta_n(X_n)} \big{\{}\E_{X_n} [ \{u(X_n + \beta_{n-1} - \beta_{n}) + \mathcal{U}_{n+1}(X_{n+1},\cdots, X_N|X_n,\beta_n)\}|X_{n-1}=x_{n-1}]\big{\}} \\
    &=&  \max_{\beta_n(X_n)} \big{\{}\E_{X_n} [ \{u(X_n + \beta_{n-1} - \beta_{n})\\
    &&  + \mathcal{U}_{n+1}(PCEV_{n+1}(X_{n+1},\cdots, X_N|X_n,\beta_n),0,\cdots,0|X_n,\beta_n\}|X_{n-1}=x_{n-1}]\big{\}}\\
    &=& \max_{\beta_n(X_n), \beta_{N-1},\beta_N=0 } \big{\{}\E_{X_n} [ \{u(X_n + \beta_{n-1} - \beta_{n})  \\
&& + u(PCEV_{n+1}(X_{n+1},\cdots, X_N|X_n,\beta_n) + \beta_n-\beta_{n+1})
+\sum_{i=n+2}^N u(\beta_{i-1}-\beta_i)\}|X_n, \beta_n, X_{n-1}=x_{n-1}]\big{\}}
\end{eqnarray*}
By induction hypothesis $PCEV_{n+1}$ is independent of $\beta_n$, using variable substitution $\beta'_n = \beta_n + PCEV_{n+1}(X_{n+1},\cdots, X_N|X_n,\beta'_n)$, the above expression becomes:
\[ \mathcal{U}_n(X_n + PCEV_{n+1}(X_{n+1},\cdots,X_N|X_n,\beta'_n), 0, \cdots, 0|x_{n-1},\beta_{n-1}) \]
(~\ref{eqn:PCEVMK}) concludes the proof of statement (2).  Statement (1) follows from  statement (2) and Lemma~\ref{lem:oneCF-PCEV}.
\endproof

Theorem \ref{theo:PCEVMK} implies that we need to compute the PCEV value recursively when the future cash flow depends on the realization of the past cash flow. This is in contrast to Theorem \ref{theo:indCash}, which states that each cash flow's PCEV can be computed independently.

\section{PCEV with Financial Hedging of a Financial Asset}

In this section we consider the uncertainty of the decision maker's cash flow  that comes from a financial security traded in the market. At the beginning of each period, the security price is realized through market trading, while its future price is uncertain depending on realization of uncertain future events. A typical example is the value of another county's currency that is traded in the currency exchange market. 

Since the future price of financial security is the only source of uncertainty and it is traded in a market, the decision maker can increase his utility by financial hedging (buying or selling the security in the market). A well functioning financial market has no arbitrage opportunities. If the financial market consists of a risky financial security and a risk-free bond and has no arbitrage, then there exits a risk-neutral measure such that all derivatives (contingent claims) on the risky security can be priced by taking expectation under this risk-neutral measure. Formally for two periods we have    
\begin{assumption}
If there is no arbitrage opportunity in the trading of the risky security and the risk free bond return is 0, then there exits a risk neutral measure $M$ such that:\\
(a) $\E_0^M [S_1|S_0] = S_0 $ \\
(b) Let $C_{01}(S_0,S_1)$ be any derivative that is written on the security at time 0 and expires at time 1.  Then, its price is $\E_0^M[C_{01}(S_0,S_1)|S_0]$ at period 0.
\end{assumption}

In this section we define financial hedging as any activity in the financial market that involves buying and selling the risky security, in combination with or without the risk-free bond trading.


\subsection{A Single Uncertain Cash Flow - One Period Problem}
We consider the simplest case of a certain cash flow 0 in period 0, and a uncertain cash flow in period 1, $X_1$, that is a function of the security's price $S_1$. The present certainty equivalent value of this cash flow $(0, X_1(S_1))$ is defined through the maximum utility that the decision maker generates in combination with financial hedging.

\begin{align} \label{eqn:maxUtiFH2Pd}
    &\mathcal{U}^N_0(0, X_{1}(S_1)|\beta_{-1},S_0) \nonumber \\
    & = \max_{\beta_0(S_0), C_1(S_0,S_1)} \E_0^N \big{[}U\big{(}\beta_{-1}-\beta_0(S_0)-\E_0^M[C_{01}(S_0,S_1)|S_0],  \nonumber \\ & X_1(S_1)+ C_{01}(S_0,S_1) + \beta_{0}(S_0)\big{)}|S_0 \big{]}
\end{align}

In Equation~\ref{eqn:maxUtiFH2Pd}, the first expectation is taken with respect to information at time 0 and natural probability of the security uncertainty, while the second expectation with respect to market risk neutral probability. To distinguish between the two different probability, we let $N$ be the superscript standing for natural probability measure. 

Similarly to ~\ref{eqn:PCEV}, the $PCEV^N_0$ with financial hedging is defined as,
\begin{align} \label{eqn:PCEV-FH2Period}
\mathcal{U}^N_0(PCEV_0^N, 0|\beta_{-1},S_0) = \mathcal{U}^N_0(0, X_1(S_1) |\beta_{-1},S_0)
\end{align}


%
%
%\begin{lemma} Under risk neutral probability measure, a single period problem has the property:
%\[\text{(a) the maximum utility is}: \;\mathcal{U}_1^M(X_1(S_1)|S_0) = u(\E^M[X_1(S_1)|S_0])\]
%\[\text{(b) the optimal financial contract is:}\; C(S_1) =\E^M[X_1(S_1)] -X_1(S_1) \]
%\end{lemma}
%\proof We proof (a) first. By definition, let $C(S_1)$ be an financial hedging contract, then
%\begin{eqnarray*}
%\mathcal{U}^M_1(X_1(S_1)|S_0) &=& \max_{C(S_1) \in \{\E^M[C(S_1)|S_0] = 0\}} \E^M [\{u(X_1(S_1)+C(S_1))\}|S_0]\\
%&\leq & u(\E^M[X_1(S_1) + C(S_1)|S_0])  \text{by concavity of $u$} \\
%&=& u(\E^M[X_1(S_1)|S_0]) \text{by property of financial contract}
%\end{eqnarray*}
%The part (b) follows form the part (a).
%\endproof
%\subsubsection{A two-period problem}


We now study whether the present certainty equivalent value depends on the probability measures, in this simple example.

Figure~\ref{assetPrice} shows a risky asset whose value evolves from a deterministic $S_0=2$ in period $0$ to an uncertain $S_1$ in period $1$. Under natural probability measure (Figure~\ref{assetPrice} (a)), it is evident that the asset price has a positive average return because the $\E_0^N (S_1|S_0) = 3.1 > 2 = S_0=\E_0^M[S_1|S_0]$.  

\begin{figure}
\includegraphics[scale=0.6]{assetPrice.pdf}\newline
\caption{a risky asset price}%
\label{assetPrice}%
\end{figure}
\bigskip

Since there are only two realizations of the asset price, for notation clarity, let
\[ x_1 = C_{01}(2,4), x_2 = C_{01}(2,1), \beta_{-1} = 0 \]
We use (~\ref{eqn:maxUtiFH2Pd}) to find PCEV, the decision variable $\beta_0(S_0)$ is redundant and can be ignored. When utility function is additive exponential, those notation reduces the problem to
\[ \max_{x_1,x_2} \{ u(-1/3 x_1 - 2/3 x_2) + p_1^Nu(4+x_1) + p_2^Nu(1+x_2)\}\]
Since the objective function is concave, verifying the first order condition suffices for finding the optimal contract,
\begin{align}
&-1/3u'(-1/3x_1-2/3 x_2) + p_1^Nu'(4+x_1) = 0\\
&-2/3u'(-1/3x_1-2/3 x_2) + p_2^Nu'(1+x_2) = 0
\end{align}
Since the utility is exponential, we have $u'(x_1) =-u(x_1)$. Thus, we have
\begin{align}
&1/3u(-1/3x_1-2/3 x_2) = p_1^Nu(4+x_1)\\
&2/3u(-1/3x_1-2/3 x_2) = p_2^Nu(1+x_2) 
\end{align}
And the maximum utility is 
\[\mathcal{U}^N_0(0, S_1|S_0,0) = 2u(-1/3x_1-2/3x_2) = 2u(-(p_1^M x_1+p_2^M x_2)) \]
These optimal contract, and consequently the corresponding maximum utility, clearly depend on the probability measure.  We first assume that the natural probability measure coincides with the risk neutral probability measure. 

{\example If the $N=M$, then $x_1 = -3$, $x_2 = 0$. }

This solution represents the decision maker shorts a call option on the asset at time 0: if the asset value turns out to be 4, he will pay 3 to the counter party; else he pays nothing. The value of this call option is $1$ under risk neutral measure. With this call option, the decision maker consumes 1\$ in period 0 and 1\$ in period 1, regardless of the realization of the risky asset price. 


What would happen if the Natural probability measure is different from the risk neutral measure:

{\example If $p^N_1 = 0.7$ and $p_2^N =0.3$, then $x_1 = -2.55$ and $x_2 = -0.40$ }

This financial transaction can be viewed as the decision maker shorts a digital call option that pays the counter party 2.15\$ if the asset value goes up to 4\$ and 0\$ otherwise at time 1. He also borrows 0.40\$. This combination of transactions, allows him to consume equally between period 0 and period 1.

It is evident from this example, the optimal financial hedging contract for the maximum utility satisfies the following property. 

\begin{align} \label{eqn:fonc-cash}
p_i^Mu(-\sum_{j=1}^2 p_j^M x_j) = p_i^N u(s_i + x_i)\quad  \forall i \in \{1,2\}
\end{align}
The left side of the equation is the multiplication of the utility in period 0 times the risk neutral probability of an event. The right side of the equation is the expected utility under the same event of the natural probability measure. The optimal consumption cash flow equalizes them.  

Under the optimal consumption cash flow, the maximum utility is,
\begin{align} \label{eqn:maxU-cash}
\mathcal{U}^N_0 (0, S_1|\beta_{-1},S_0)  = 2u(-\sum_{j=1}^2 p_j^M x_j) 
\end{align}
This maximum utility implies that in each period 0 and 1, the utility is the same. 

\subsubsection{ Present Certainty Equivalent Value}
In this section we show that independent of the probability measure, the present certainty equivalent value of the cash flow $(0, S_1)$ is  $\E_0^M [S_1|S_0]$, the present expected value of $S_1$ under risk-neutral measure. We start with the study of the maximum utility under a present certainty equivalent value under a natural probability measure.

By (\ref{eqn:maxUtiFH2Pd}), the left hand side of (\ref{eqn:PCEV-FH2Period}) becomes
\begin{align*}
&\mathcal{U}^N_0 (PCEV_0^N, 0|\beta_{-1},S_0)  \nonumber \\&= \max_{\beta_0(S_0), C_{01}(S_0,S_1)} \E_0^N [U(\beta_{-1} + PCEV_0^N - \beta_0(S_0) - E_0^M[C_{01}(S_0,S_1)|S_0], \\&C_{01}(S_0,S_1) + \beta_0(S_0))|S_0]
\end{align*}

Now we solve the above optimization problem similarly for the example given in Figure~\ref{assetPrice}. Since there are only two realizations of the asset price, for notation clarity, let
\[ y_1 = C_{01}(2,4), y_2 = C_{01}(2,1), \beta_{-1} = 0 \]
And similarly $\beta_0$ is a redundant decision variable, thus the optimization problem becomes,
\[ \max_{y_1,y_2} \{ u(PCEV_0^N-\sum_{j=1}^2p_j^M y_j) + p_1^Nu(y_1) + p_2^Nu(y_2)\}\]
The first order optimality condition is,
\begin{align} \label{eqn:fonc-cerEq}
p_i^M u(PCEV_0^N - \sum_{j=1}^2 p_j^M y_j) = p_i^N u(y_i) \quad \forall i\in \{1,2\} 
\end{align}
And the maximum utility is
\begin{align} \label{eqn:maxU-cerEq}
\mathcal{U}^N_0 (PCEV_0^N, 0|\beta_{-1},S_0)  =2u(PCEV_0^N-\sum_{j=1}^2 p_j^My_j)
\end{align}

By definition of the certainty equivalent, the maximum utility must equal. Equations~(\ref{eqn:maxU-cash}), (\ref{eqn:maxU-cerEq}), (\ref{eqn:fonc-cash}) and (\ref{eqn:fonc-cerEq}) imply:
\begin{align}
PCEV_0^N - \sum_{j=1}^2 p_j^My_j &= -\sum_{j=1}^2 p_j^Mx_j \\
s_i + x_i &= y_i \quad \forall i \in \{1,2\} \label{eqn:s_i}
\end{align}

Therefore we have
\[PCEV_0^N = \sum_{j=1}^2 p_j^M s_j  = \E_0^M[S_1|S_0]\]
{\lemma If the cash flow is $(0,S_1)$ where $S_1$ is the value of a tradable market security at time 1, then the present certainty equivalent value of the cash flow is its expected value under risk neutral measure, independent of the natural probability measure.}


In summary, while the optimal trading strategy to maximize the utility depends on the probability measures, the certainty equivalent value depends on the risk neutral measures only. My next work is to extend the results to a more general cash flow.

\subsubsection{Cash flow as an arbitrary function of the security}
In this section we extend the results of the previous section to cash flow as a function of the security, let the general cash flow in period 0 and 1 be
\[(0, X_1(S_1))\]

If we allow that in Equation~(\ref{eqn:s_i}) $s_i$ is replaced by $X_1(s_i)$, then the following Lemma follows from the same analysis as in the previous subsection. 
{\lemma The present certainty equivalent value of cash flow $(0,X_1(S_1))$ is independent of the probability measure and 
\[PCEV_0^N(0,X_1(S_1)) = \E_0^M[X_1(S_1)|S_0] \]
}
\proof We start with definition Equation~\ref{eqn:maxUtiFH2Pd}. Since the contract contains the bond trading, the bond decision is redundant, the maximum utility is reduced to 
\begin{align*}
    &\mathcal{U}^N_0(0, X_{1}(S_1)|\beta_{-1},S_0) \nonumber \\
    & = \max_{ C_{01}(S_0,S_1)} \E_0^N \big{[}U\big{(}\beta_{-1}-\E_0^M[C_{01}(S_0,S_1)|S_0],   X_1(S_1)+ C_{01}(S_0,S_1) \big{)}|S_0 \big{]} \\
    & = \max_{ C'_{01}(S_0,S_1)} \E_0^N \big{[}U\big{(}\beta_{-1}-\E_0^M[C'_{01}(S_0,S_1)-X_1(S_1) |S_0],   C'_{01}(S_0,S_1) \big{)}|S_0 \big{]} \\
    & = \mathcal{U}^N_0(\E_0^M[ X_{1}(S_1)|S_0], 0)|\beta_{-1},S_0)
\end{align*}
Where the second equality is due to variable substitution $C'_{01}(S_0,S_1) = X_1(S_1) + C_{01}(S_0,S_1)$ and the last equality is due to the definition of the maximum utility. The proof is concluded by Equation 
~(\ref{eqn:PCEV-FH2Period}). 
\endproof

This proof sheds some lights on how to generalize the result into more general cash flow and time periods. The proof makes use of only the property of the price of derivatives under risk neutral measure and the definition of the present certainty equivalent value, it does not require any property of utility function. Thus this result is very general, it is true for risk-averse decision maker, risk-seeking decision maker, and any consumption preference of the decision maker. 

To understand the intuition behind this proof, we first note that by definition the present certainty equivalent value of a stream of future cash flows is an amount of cash at present time that generates the same maximum utility as the stream of future cash flows. In another words, the consumer is indifferent between the present certainty equivalent value now and the stream of future cash flows. It implies that the present certainty equivalent value is the cash value of the stream of future cash flows. 

Second when the stream of future cash flows are the function of the underlying security only, they are contingent claims on the underlying security, thus they can be replicated using a portfolio of derivatives of the underlying  security. Thus the present certainty equivalent value (the cash price) of the stream of future cash flows equals the cash value of its replicating portfolio of derivatives. 

Thirdly the cash value of each derivative is its expected value under risk-neutral measure under complete market. Thus the cash value of each derivative is independent of probability measure. The cash value of the replicating portfolio of derivatives is independent of probability measure.

Thus the certainty equivalent value of the stream of future cash flows is independent of the probability measure because it equals the cash value of the replicating portfolio of derivatives. 

We now make give a second proof using the property of the property of the consumption cash flow at the maximum utility.

{\assumption \label{ass:consumpUnique} 
The resulted  optimal consumption cash flow of problem in equation~(\ref{eqn:maxUti}) is unique, i.e.,
\begin{align} 
\E U(c_0, \cdots, c_N)  = \E U(c'_0, \cdots, c'_N)  \quad  \mbox{iff} \quad c'_i = c_i \; \forall i \in \{0, \cdots, N \} 
\end{align}
where the $c_i$ and $c'_i$ are optimal consumption cash flow for any income cash flow. 
}

{\lemma If $\mathcal{U}(0,S_1|S_0) = \mathcal{U}(PCEV,0|S_0)$ and let the optimal derivatives  be
\begin{align}
C^*_{01}(S_0,S_1) &= \arg\max_{C_{01}(S_0,S_1)} \E^N\big{[} U\big{(}-\E^M_0[C_{01}(S_0,S_1)|S_0], S_1 + C_{01}(S_0,S_1) \big{)}|S_0\big{]}\\
C'_{01}(S_0,S_1) &= \arg\max_{C_{01}(S_0,S_1)} \E^N\big{[} U\big{(}PCEV-\E^M_0[C_{01}(S_0,S_1)|S_0], C_{01}(S_0,S_1) \big{)}|S_0\big{]}
\end{align}
then  
\begin{align}
& (a) \; C'_{01}(S_0,S_1) = C^*_{01}(S_0,S_1) + S_1 \\
& (b) \; PCEV = E_0^M[S_1|S_0]
\end{align}
}
\proof Since the maximum utilities are equal at the optimal consumption cash flow, Assumption~\ref{ass:consumpUnique} implies the second period consumption cash flow must equal under both maximization problem, i.e., part (a) is true.

Since the first period consumption cash flow also must be equal, we have
\[PCEV = \E_0^M[ C'_{01}(S_0,S_1) - C^*_{01}(S_0,S_1) |S_0] \]
Combining it with part (a) completes proof of part (b).
\endproof




This intuition can be further understood by an argument similar to the arbitrage pricing. Suppose the PCEV of a cash flow is different from the price of the cash flow's replication derivatives, then the maximum utilities generated by the PCEV and the cash flow will be different, which contradicts to the definition of the PCEV. 

To show this, we first consider that the PCEV is strictly greater than the price. In this case, we can use this PCEV to create another cash flow that has more cash in period 0 than the cash flow and in all other periods has identical cash. Thus this new cash flow must generate more utility than the original cash flow. However, this new cash flow must generate no more utility than the PCEV because it is one feasible income cash flow generated by the PCEV. Thus, the PCEV generates more utility than the original cash flow, which contradict the definition of the PCEV. 

Second if the PCEV is strictly smaller than the price of the replicating derivative of portfolio of the cash flow. Then we can transform the replicating portfolio into its price at time 0 by selling the derivatives in market. Since the price is strictly greater than the PCEV, this price must generate strictly more utility than the PCEV, again this contradicts the definition of PCEV. 

In summary when PCEV is differ from the price of the replicating derivatives of the cash flow and we trade the derivatives in the market, it causes contradiction to the definition of the PCEV. Thus, the PCEV of a cash flow is the price of its replicating derivatives. 

\subsubsection{PCEV of an income cash flow $(0,C_1(S_1))$ where the $S_1$ has three realizations}

We now extend the PCEV result to a multiple security price realization for a two period problem. We can trade the derivatives of the security to maximize our utility. We first define the underlying security and its derivatives. 


Let the security price in period 0 is $S_0 =s_0$ and in period 1 have $3$ possible realizations.  The probability of realization under risk neutral measure is,
\[p_i^M = \text {prob}^M (S_1 = s_i) \; \forall i \in \{1,2,3\} \]
This must satisfy,
\[ \sum_{i=1}^3 p_i^M s_i = s_0 \]
Let the natural probability of the security price realization to be
\[p_i^N = \text{prob}^N(S_1 = s_i) \; \forall i \in \{1,2,3 \} \]
Under the natural probability measure, the expected value of the security price in period 1 does not have to be its price in period 0. 
 
 A derivative of the security is defined by its payoff at period 1
 \[D =d_i \; \text{with probability} \; p_i^N, \quad \forall i\in\{1,2,3\}\]
 and its price at period 0 is its expected value under risk neutral measure
 \[\E_0^M[D] = \sum_{i=1}^3[p_i^M d_i] \]
 
 
 Now since the cash flow in the period 1 is a function of the security price, thus it has three realizations corresponding to the three realizations of the security price in period 1. 
 \[ \forall i \in \{1, 2, 3\} \; \text{prob}^N(C_1 = c_i) = p_i^N \]
 
 
Since the derivatives are available in the market, the maximum utility generated by cash flow $(0,C_1(S_1))$ is
\begin{align}
\mathcal{U}(0,C_1(S_1))&= \max_{D}\big{\{} E_0^N [U(0-\E_0^M[D], C_1(S_1) + D)]\big{ \}}\nonumber \\
& = \max_{d_i, \forall i} \big{\{}\sum_{i=1}^3 [p_i^N U(-\sum_{i=1}^3 p_i^M d_i, c_i + d_i)]\big{ \}} \label{eqn:maxUti3PeriodFH}
\end{align}

{\lemma If $D^*$ is optimal to Equation~(\ref{eqn:maxUti3PeriodFH}), then
\[\bar{D}^* = D^* + C_1(S_1) \] 
is optimal to 
\[ \max_{\bar{D}} \big{\{} E_0^N [U(\E_0^M[C_1(S_1)]-\bar{D},  \bar{D} )]\big{ \}} \]
and
\[PCEV(0, C_1(S_1)) = \E_0^M[C_1(S_1)]\]
} 

\proof Using variable substitution $\bar{D} = D + C_1(S_1)$, the problem in Equation~(\ref{eqn:maxUti3PeriodFH}) is equivalent to
\[\max_{\bar{d}_i, \forall i} \big{\{}\sum_{j=1}^3 [p_j^N U(\E_0^M[C_1(S_1)]-\sum_{i=1}^3 p_i^M \bar{d}_i, \bar{d}_i)]\big{ \}} \]
Thus if $D^*$ is optimal to the original problem, then $\bar{D}^*$ is optimal to the above problem. Furthermore, the maximum utilities of these two problems are equal, i.e.,
\[\mathcal{U}(0, C_1(S_1)) = \mathcal{U} (\E_0^M[C_1(S_1)],0) \]
This completes the proof of the PCEV. 
\endproof


\include{multiplePeriodFH}

\section{Income Cash Flow  with Two Random Sources}
In this section we consider the MRN faces two random sources. One source is from the uncertainty of demand and the other is from the exchange rate.  Let\\
$\alpha_i$: the production strategy in period $i$\\
$D_i$: the product demand in period $i$\\
$S_i$: the exchange rate value in period $i$ \\
We will assume that the demand distribution and the exchange rate distribution are independent of each of other. 


Formally, the uncertain income cash flows are only functions of the operations decisions and the demand and the exchange rate.
\begin{align}
X_n(\alpha_n, D_n,S_n), \cdots, X_i(\alpha_i, D_i,S_i), \cdots X_N(\alpha_N,D_N,S_N)
\end{align}
Since the MRN's goal is to maximize his PCEV, We need to transform the income cash flow into consumption cash flow through financial instruments. Thus we need to discuss the MRN's consumption decisions. 

\subsection{PCEV with Bond Investment}
In this section we make the assumption that the decision maker uses only bond to change the income cash flow into consumption cash flow. MRN stops consumption after period $N$, thus the bond investment in period $N$ is 0. Therefore, for a given production strategy, the MRN's maximum utility is
\begin{align}
& \mathcal{U}_n(X_n(\alpha_n, D_n,S_n), \cdots, x_N(\alpha_N,D_N,S_N)|\beta_{n-1}) =  \max_{\beta_n, \cdots, \beta_{N-1},\beta_N =0}\E[U(X_n(\alpha_n,D_n,S_n) +\beta_{n-1}-\beta_n, \nonumber \\
&\cdots, X_i(\alpha_i,D_i,S_i) + \beta_{i-1}-\beta_i, \cdots, X_N(\alpha_N,D_N,S_N)+\beta_{N-1} - \beta_{N} )]
\end{align} 
The corresponding definition of PCEV then is
\begin{align}
\mathcal{U}_n(PCEV_n, 0, \cdots, 0|\beta_{n-1} ) = \mathcal{U}_n(X_n(\alpha_n, D_n,S_n), \cdots, x_N(\alpha_N,D_N,S_N)|\beta_{n-1})
\end{align}
To build our intuition on the optimal operation strategy and the optimal consumption plan, we start with the simplest case. Since a single period problem has no bond investment decisions, it does not capture the key feature of our model. Our simplest case with bond investment thus is a two period problem. Assuming initial wealth is 0 and the utility function is additive and stationary, we have
\begin{align*}
\max_{\beta_1} \E[u(X_1(\alpha_1, D_1, S_1)-\beta_1) + u( \beta_1)]
\end{align*} 
It is evident that the optimal $\beta_1$ equalizes the two period cash flows, thus the MRN's utility as a function of the operation decision becomes
\[2\E [u(X_1(\alpha_1,D_1,S_1)/2)]\]
Now the optimal decision to maximize the decision maker's utility is
\[ \max_{\alpha_1} \{2\E [u(X_1(\alpha_1,D_1,S_1)/2)]\} \]
When the operations decision is made after the realization of demand and exchange rate, the problem becomes
\begin{align*}
 2\E [\max_{\alpha_1} \{u(X_1(\alpha_1,D_1,S_1)/2)\}]
\end{align*}
If the utility function is increasing concave, we have
\begin{align*}
2\E [u(\max_{\alpha_1} \{(X_1(\alpha_1,D_1,S_1)\}/2)]
\end{align*}
Which implies that the whole problem can be solved by finding the optimal operations decisions in each period first and then using financial instruments to make the optimal consumption decision. 
%{\definition}
We now generalize this idea into the MRN's operation and consumption problem for multiple periods.

\subsection{MRN's Income Cash Flow}
In each period, MRN's income cash flow is determined by his operation decisions in each period and the uncertain demand and exchange rate. To express the MRN's one period cash flow, we use the following notation.
\begin{itemize}
    \item $h$ and $o$ superscripts of home and overseas markets or facilities, respectively,
    \item $d^j$ market $j$ demand, $\bm{d} = \{d^h, d^o \} $,
    \item $k^j$ capacity of facility $j$,   $\bm{k} = \{k^h,k^o\}$,
    \item $z^{j}$ quantity produced in facility $j$ for market $j$, $\bm{z} = \{z^{h},z^o\}$,
    \item $x^{j}$ quantity produced in the other facility and transshipped to market $j$, $\bm{x} = \{x^h,x^o\}$,
    \item $i$:  subscript index of the period number, counting backwards,
\end{itemize} 
Thus, the income cash flow in period $i$ is
\begin{align}
J_i(\bm{z}_i,\bm{x}_i, s_i) =  (p-c)z_i^{d}+ (p-cs_i-t)x_i^{d} + s_i(p-c)z_i^{o}+(s_i p-c-t)x_i^{o} 
\end{align}
where the production quantity and transshipment are subject to the demand and capacity constraint. For brevity, we suppress the index $i$. The operation decision feasible set in each period is a function of the demand and the capacity.
\begin{align}
A(\bm{k},\bm{d}) = \{\bm{z} + \bm{x} \leq \bm{d}, z^j + x^{l \neq j} \leq k^j \; \forall j \in\{h,o\}\; \forall l \in \{h,0\} , \bm{z}\geq 0, \bm{x} \geq 0\}
\end{align}
MRN's objective then is to maximize the present certainty equivalent value of his income cash flows.
\begin{align}
\max_{(\bm{z}_i\bm{x}_i) \in A(\bm{k},\bm{d}_i), \forall i}\{PCEV_n(J_n(\bm{z}_n,\bm{x}_n,s_n),\cdots, J_i(\bm{z}_i,\bm{x}_i,s_i), \cdots, J_N(\bm{z}_N,\bm{x}_N,s_N))\} \end{align}
Since the present certainty equivalent value of the MRN is a determined by his utility function and his consumption cash flows, we must consider his trading strategy in the financial market. We first consider the MRN trades only bond to optimize his consumption.

\subsection{MRN's PCEV Maximization with Bond Investment}

When MRN uses bond trading to optimize his consumption, his present certainty equivalent value maximization problem becomes
\begin{align*}
\max_{(\bm{z}_i\bm{x}_i) \in A(\bm{k},\bm{d}_i) \forall i}\big{\{} \max_{\beta_i,\forall i<N, \beta_N=0}\{&\E[U(J_n(\bm{z}_n,\bm{x}_n,S_n)+\beta_n-\beta_{n-1}, \cdots,\\ & J_i(\bm{z}_i,\bm{x}_i,S_i)+ \beta_{i-1} - \beta_i, \cdots,J_N(\bm{z}_N,\bm{x}_N,S_N)+\beta_{N-1} ) ] \}\big{\}}
\end{align*}
If the utility function is additive, the above problem becomes
\begin{align*}
&\max_{(\bm{z}_i\bm{x}_i) \in A(\bm{k},\bm{d}_i) \forall i, \beta_i,\forall i<N, \beta_N=0}\big{\{} \E[\sum_{i=1}^N u_i( J_i(\bm{z}_i,\bm{x}_i,S_i)+ \beta_{i-1} - \beta_i) ] \big{\}} \\
& =\max_{\beta_i,\forall i<N, \beta_N=0}\big{\{} \E [\sum_{i=1}^N \max_{(\bm{z}_i\bm{x}_i) \in A(\bm{k},\bm{d}_i)}\{u_i(J_i(\bm{z}_i,\bm{x}_i,S_i)+ \beta_{i-1} - \beta_i)\} ] \big{\}} 
\end{align*}



\newpage

%\include{zenAndSobel}

\end{document}